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Hilbert Transform

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The Hilbert transform (and its inverse) are the integral transform

g(y)=H[f(x)]=1/piPVint_(-infty)^infty(f(x)dx)/(x-y)
(1)
f(x)=H^(-1)[g(y)]=-1/piPVint_(-infty)^infty(g(y)dy)/(y-x),
(2)

where the Cauchy principal value is taken in each of the integrals. The Hilbert transform is an improper integral.

In the following table, Pi(x) is the rectangle function, sinc(x) is the sinc function, delta(x) is the delta function, AdjustmentBox[I, BoxMargins -> {{0.13913, -0.13913}, {-0.5, 0.5}}]I(x) and AdjustmentBox[I, BoxMargins -> {{0.101266, -0.101266}, {0.375, -0.375}}, BoxBaselineShift -> -0.375]AdjustmentBox[I, BoxMargins -> {{0, 0}, {-0.25, 0.25}}, BoxBaselineShift -> 0.25](x) are impulse symbols, and _1F_1(a;b;x) is a confluent hypergeometric function of the first kind.

f(x)g(y)
sinxcosy
cosx-siny
(sinx)/x(cosy-1)/y
Pi(x)1/piln|(y-1/2)/(y+1/2)|
1/(1+x^2)-y/(1+y^2)
sinc^'(x)(1-cosy-ysiny)/(y^2)
delta(x)-1/(piy)
AdjustmentBox[I, BoxMargins -> {{0.13913, -0.13913}, {-0.5, 0.5}}]I(x)y/(pi(1/4-y^2))
AdjustmentBox[I, BoxMargins -> {{0.101266, -0.101266}, {0.375, -0.375}}, BoxBaselineShift -> -0.375]AdjustmentBox[I, BoxMargins -> {{0, 0}, {-0.25, 0.25}}, BoxBaselineShift -> 0.25](x)-1/(2pi(1/4-y^2))
e^(-x^2)-e^(-y^2)erfi(y)

See also

Abel Transform, Fourier Transform, Improper Integral, Integral Transform, Titchmarsh Theorem, Wiener-Lee Transform

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References

Bracewell, R. "The Hilbert Transform." The Fourier Transform and Its Applications, 3rd ed. New York: McGraw-Hill, pp. 267-272, 1999.Papoulis, A. "Hilbert Transforms." The Fourier Integral and Its Applications. New York: McGraw-Hill, pp. 198-201, 1962.

Referenced on Wolfram|Alpha

Hilbert Transform

Cite this as:

Weisstein, Eric W. "Hilbert Transform." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/HilbertTransform.html

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